Friday, July 31, 2015

The dataflow engine I gave in my last post can be seen as an
implementation of self-adjusting computation,
in the style of Acar, Blelloch and Harper's original POPL 2002
paper Adaptive Functional Programming.
(Since then, state of the art implementation techniques have
improved a lot, so don't take my post as indicative of what
modern libraries do.)

Many people have seen resemblances between self-adjusting computation
and functional reactive programming --- a good example of this is Jake
Donham's Froc library for
Ocaml. Originally, I was one of those people, but that's no
longer true: I think SAC and FRP are completely orthogonal.

I now think that FRP libraries can be very minimalistic --- my ICFP
2013 paper Higher-Order Reactive Programming without Spacetime Leaks
gives a type system, implementation, and correctness proof for
an FRP language with full support for higher order constructions
like higher-order functions and streams of streams, while at the
same time statically ruling out space and time leaks.

The key idea is to distinguish between stable values (like
ints and bools) whose representation doesn't change over time from
dynamic values (like streams) whose representation is
time-varying. Stable values are the usual datatypes, and can be used
whenever we like. But dynamic values have a scheduling constraint: we
can only use them at certain times. For example, with a stream, we
want to look at the head at time 0, the head of the tail at time 1,
the head of the tail of the tail at time 2, and so on. It's a
mistake to look at the head of the tail of a stream at time 0, because
that value might not be available yet.

With an appropriate type discipline, it's possible to ensure
scheduling correctness statically, but unfortunately many people are
put off by modal types and Kripke logical relations. This is a shame,
because the payoff of all this is that the implementation strategy is
super-simple -- we can just use plain-vanilla lazy evaluation to
implement FRP.

Recently, though, I've figured out how to embed this kind of FRP
library into standard functional languages like Ocaml. Since we can't
define modal type operators in standard functional languages, we have
to give up some static assurance, and replace the static
checks of time-correctness with dynamic checks, but we are
still able to rule out space leaks by construction, and still get a
runtime error if we mis-schedule a program. Essentially, we can
replace type checking with contract checking.
As usual, you can find the code on Github here.

The NEXT signature introduces a single type constructor 'a
t, which can be thought of as the type of computations which
are scheduled to be evaluated on precisely the next tick of the
clock. The elements of 'a t are dynamic in the
sense that I mention above: we are only permitted to evaluate it on
the next tick of the clock, and evaluating it at any other time is an
error.

To model this kind of error, we also have a Timing_error
exception, which signals an error whose first argument contains the
time a thunk was scheduled to be evaluated, and whose second argument
contains the actual time.

Elements of 'a t are the only primitive way to create
dynamic values -- other values (like function closures) can be
dynamic, but only if they end up capturing a next-step thunk.

The delay function lets us create a next value from a
thunk, and the map function maps a function over a thunk.
The zip and unzip are used for pairing,
and the $ operation is the McBride/Paterson idiomatic
application operator. (Technically, it's derivable from zip,
but it's easiest to throw it in to the basic API.)

The fix operation is the one that really makes
reactive programming possible -- it says that guarded recursion
is allowed. So if we have a function which takes an 'a next
and returns an 'a, then we can take a fixed point. This
fixed point will always never block the event loop, because its
type ensures that we always delay by a tick before making a recursive
call.

This raw interface is, honestly, not so useful as is, but the slightly
miraculous fact is that this is the complete API we need to build all
the higher-level abstractions --- like events and streams --- that we
need to do real reactive programming.

Now, let's see what an implementation of this library could look like.

moduleNext :NEXT =structlettime =ref 0

We can keep track of the current time in a reference cell.

type'a t ={time:int;mutablecode:'a Lazy.t}

The type of a thunk is a record consisting of a lazy thunk, and the time when it is
safe to force it.

types =Hide:'a t -> s letthunks :s list ref=ref[]

We also have a list that stores all of the references that we've
allocated. We'll use this list to enforce space-safety, by mutating
any thunk that gets too old.

When we create a thunk with the delay function, we are
creating a thunk to be forced on the next time tick. So we can dererefence
time in order to find out the current time, and add 1 to
get the scheduled execution time for the thunk. We also add it to the
list thunks, so that we can remember that we created it.

Forcing a thunk just forces the code thunk, if the current time
matches the scheduled time for the thunk. Otherwise, we raise a
Timing_error. Note that memoization is handled by Ocaml's
built-in 'a Lazy.t type.

The tick function advances time by doing two things. First,
it increments the current time.
Then, it filters the list of thunks using the cleanup
function, which does two things. First, cleanup returns
true if its argument is older than the current time. As a result, we
only retain thunks in thunks which can be forced now or in
in the future.

Second, if the argument thunk to cleanup is old, it
replaces the code body with an assertion failure, since no
time-correct program should ever force this thunk. Updating the code
ensures that by construction next-step thunks always lose
their reference to their data once they age out, because every
thunk is placed onto thunks when it is created, and
when the clock is ticked past its time, it is guaranteed to drop
its references to its data.

This guarantees that spacetime leaks are impossible, since
we dynamically zero out any thunks that get too old! So here we see
how essential data abstraction is for imperative programming,
and not just functional programming.

As you can see, the implementation of the Next library is
pretty straightforward. The only mildly clever thing we do is to keep
track of the next-tick computations so we can null them out when they
get too old.

You should be wondering now how we can actually write reactive
programs, when the primitive the API provides only lets you schedule a
computation to run on the next tick, and that's it. The answer is
datatype declarations. Now that we have a type that lets
us talk about time, We can re-use our host language's facility to
define types which say more interesting things about time.

Let's start with the classic datatype of functional reactive
programming: streams. Streams are a kind of lazy sequence, which
recursively give you a value now, and a stream starting tomorrow,
thereby giving you a value on every time step.

We give a simple signature for streams above. They are
a datatype exactly following the English description above,
as well as a collection of accessor and constructor functions,
like head, tail, map,
unfold and so on. All of these have pretty much
the expected types.

The only difference from the usual stream types is that sometimes
we need a Next.t to tell us when a value
needs to be available. Now let's look at the implementation.

The unfold function uses a function f and an initial
seed value to incrementally produce a sequence of values. This is
exactly like the usual unfold, except we have to use the applicative
interface to the 'a Next.t type to apply the function.

zip and unzip work about the way we'd expect,
in that we use Next.zip and Next.unzip to
put together and take apart delayed pairs to build the ability to
put together and take apart streams.

This is all very nice, but the real power of giving a reactive API based on a
next-step type is that we can build types which aren't streams. For
example, let's give a datatype of events, which is the type
of values which will become available at some point in the future,
but we don't know exactly when.

We represent this with a datatype 'a event, which has
two constructors. We say that an 'a event is either a value
of type 'a available Now, or we have to
Wait to get another event tomorrow. So this is a
single value of type 'a that could come at any time ---
and we don't know when!

Events also form a monad, which corresponds to the
ability to sequence promises or futures in the promises libraries
you'll find in Javascript or Scala. (The bind here
is a bit like the code promise.then() method in JS.

The really cool thing is that we can also join on two events to
wait for the first one to complete! This can be extended to lists
of events, if desired, but the pattern is easiest to see in the
binary case.

Of course, here's a small example of how you can actually put
this together to actually run a program. The run
function gives an event loop that runs for k
steps and halts, and prints out the first k
elements of the stream it gets passed as an argument.

Wednesday, July 22, 2015

My friend Lindsey Kuper
recently remarked on Twitter that spreadsheets were commonly
understood to be the most widely used dataflow programming model, and
asked if there was a simple implementation of them.

As chance would have it, this was one of the subjects of my thesis
work -- as part of it, I wrote and proved the correctness of a small
dataflow programming library. This program has always been one of my
favorite little higher-order imperative programs, and in this post
I'll walk through the implementation. (You
can find
the code here.)

As for the proof, you can look at this
TLDI paper for some idea of the complexities involved. These days
it could all be done more simply, but the pressure of proving
everything correct did have a very salutary effect in keeping
everything as simple as possible.

The basic idea of a spreadsheet (or other dataflow engine) is that
you have a collection of places called cells, each of which
contains an expression. An expression is basically a small
program, which has the special ability to ask other cells what their
value is. The reason cells are interesting is because they do
memoization: if you ask a cell for its value twice, it will only
evaluate its expression the first time. Furthermore, it's also
possible for the user to modify the expression a cell contains (though
we don't want cells to modify their code as they execute).

So let's turn this into code. I'll use Ocaml, because ML modules make
describing the interface particularly pretty, but it should all translate
into Scala or Haskell easily enough. In particular, we'll start by
giving a module signature writing down the interace.

moduletypeCELL =sig

We start by declaring two abstract types, the type 'a cell of cells containing a value of type 'a, and the type 'a exp of expressions returning a value of type 'a.

type'a celltype'a exp

Now, the trick we are using in implementing expressions is that we treat them as a
monadic type. By re-using our host language as the language of terms that lives
inside of a cell, we don't have to implement parsers or interpreters or
anything like that. This is a familiar trick to Haskell programmers, but it's
still a good trick! So we first give the monadic bind and return operators:

valreturn :'a ->'a expval(>>=):'a exp ->('a ->'b exp)->'b exp

And then we can specify the two operations that are unique to our monadic DSL: reading
a cell (which we call get), and creating a new cell (which we call cell). It's a bit unusual to be able to create new cells as a program executes, but it's rather handy.

valcell :'a exp ->'a cell exp valget :'a cell ->'a exp

Aside from that, there are no other operations in the monadic expression DSL. Now we can give the operations that don't live in the monad. First is the update operation, which modifies the contents of a cell. This should not be called from within an 'a exp terms --- in Haskell, that might be enforced by giving update an IO type.

valset :'a cell ->'a exp -> unit

Finally, there's the run operation, which we use to run an expression. This is useful mainly for looking at the values of cells from the outside.

valrun :'a exp ->'a end

Now, we can move on to the implementation.

moduleCell :CELL =struct

The implementation of cells is at the heart of the dataflow engine, and is worth
discussing in detail. A cell is a record with five fields:

The code field of this record is the pointer to the expression that
the cell contains. This field is mutable because we can alter the contents of a
cell!

The value field is an option type, which is None
if the cell has not been evaluated yet, and Some v if the code had
evaluated to v.

The reads field is a list containing all of the cells that were
read when the code in the code field was executed. If the cell
hasn't been evaluated yet, then this is the empty list.

The observers field is a list containing all of the cells that
have read this cell when they were evaluated. So the reads field
lists all the cells this cell depends on, and the observers field
lists all the cells which depend on this cell. If this cell hasn't been evaluated
yet, then observers will of course be empty.

The id contains an integer which is the unique id of each cell.

Both reads and observers store lists of dependent cells,
and dependent cells can be of any type. In order to build a list of heterogenous
cells, we need to introduce a type ecell, which just
hides the cell type under an existential (using Ocaml's new GADT syntax):

andecell =Pack:'a cell -> ecell

We can now also give the concrete type of expressions. We define an element
of expression type 'a exp to be a thunk, which when forced
returns (a) a value of type 'a, and (b) the list of cells that
it read while evaluating:

and'a exp = unit ->('a * ecell list)

Next, let's define a couple of helper functions. The id function just returns the id of an existentially packed ecell, and the union function merges two lists of ecells while removing duplicates.

The return function just produces a thunk which returns a value and an empty list of read dependencies, and the monadic bind (>>=) sequentially composes two computations, and returns the union of their read dependencies.

To implement the cell operator, we need a source of fresh id's. So
we create an integer reference, and new_id bumps the counter before
returning a fresh id.

letr =ref 0
letnew_id()= incr r;!r

Now we can implement cell. This function takes an
expression exp, and uses new_id to create a
unique id for a cell, and then intializes a cell with the appropriate
values -- the code field is exp,
the value field is None (because the cell is
created in an unevaluated state), and the reads
and observers fields are empty (because the cell is
unevaluated), and the id is set to the value we
generated.

This is returned with an empty list of read dependencies because
we didn't read anything to construct a fresh cell!

To read a cell, we need to implement the get operation. This works a bit like memoization. First, we check to see if the value field already has a value. If it does, then we can return that. If it is None, then we have a bit more work to do.

First, we have to evaluate the expression in the code field, which returns a value v and a list of read dependencies ds. We can update the value field to Some v, and then set the reads field to ds. Then, we add this cell to the observers field of every read dependency in ds, because this cell is observing them now.

Finally, we return the value v as well as a list containing the current
cell (which is the only dependency of reading the cell).

This concludes the implementation of the monadic expression language, but our API also includes an operation to modify the code in a cell. This requires more code than just updating a field -- we have to invalidate everything which depends on the cell, too. So we need some helper functions to do that.

The first helper is remove_observer o o'. This removes the cell o from the observers field of o'. It does this by comparing the id field (which was in fact put in for this very purpose).

This function is used to implement invalidate, which takes a cell, marks it as invalid, and then marks everything which transitively depends on it invalid too. It does this by we saving the reads and observers fields into
the variables rs and os. Then, it marks the current cell as invalid by setting the value field to None, and setting the observers and reads fields to the empty list. Then, it removes the current cell from the observers list of every cell in the old read set rs, and then it calls invalidate recursively on every observer in os.

This then makes it easy to implement set -- we just update the code, and then invalidate the cell (since the memoized value is no longer valid).

letset c exp =
c.code <- exp;
invalidate (Pack c)

Finally, we can implement the run function by forcing the thunk and throwing away the read dependencies.

letrun cmd = fst (cmd ())end

That's pretty much it. I think it's quite pleasant to see how
little code it takes to implement such an engine.

One thing I like about this program is that it also shows off how
gc-unfriendly dataflow is: we track dependencies in both directions,
and as a result the whole graph is always reachable. As a result, the
usual gc heuristis will collect nothing as long as anything is
reachable. You can fix the problem by using weak references to the
observers, but weak references are also horribly gc-unfriendly
(usually there's a traversal of every weak reference on every
collection).

So I think it's very interesting that there are a natural class of
programs for which the reachability heuristic just doesn't work, and
this indicates that some interesting semantics remains to be done to
explain what the correct memory management strategy for these kinds of
programs is.